THE GREAT ALGOBIST told "him in how many ways any given integer may be expressed as such a sum.* For those whose tastes are more practical we may cite Jacobi's work in dynamics. In this subject, of fundamental importance in both applied science and mathematical physics, Jacobi made the first significant advance beyond Lagrange and Hamilton. Readers acquainted with quantum mechanics will recall the important part played hi some presentations of that revolutionary theory by the Hamilton-Jacobi equation. His work in differential equations began a new era. In algebra, to mention only one thing of many, Jacobi cast the theory of determinants into the simple form now familiar to every student in a second course of school algebra. To the Newton-Laplace-Lagrange theory of attraction Jacobi made substantial contributions by his beautiful investi- gations on the functions which recur repeatedly in that theory and by applications of elliptic and Abelian functions to the attraction of ellipsoids. Of a far higher order of originality is his great discovery in Abelian functions. Such functions arise in the inversion of an Abelian integral, in the same way that the elliptic functions arise from the inversion of an elliptic integral. (The technical terms were noted earlier hi this chapter.) Here he had nothing to guide him, and for long he wandered lost in a maze that had no clue. The appropriate inverse functions in the simplest case are functions of two variables having Jour periods; in the general case the functions have n variables and 2n periods; the elliptic functions correspond to n = 1. This discovery was to nineteenth-century analysis what Columbus' discovery of America was to fifteenth-century geography. Jacobi did not suffer an early death from overwork, as his lazier friends predicted that he would, but from smallpox (18 February 1851) in his forty-seventh year. In taking leave of this large-minded man we may quote his retort to the great French mathematical physicist Fourier, who had reproached * If n is odd, the number of ways is 8 times the sum of all the divisors of n (1 and n included); if n is even, the number of ways is 24* times the sum of all the odd divisors of n. 371